In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space...

In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and...

complex vector bundle is canonically oriented; in particular, one can take its Euler class. A complex vector bundle is a holomorphic vector bundle if X {\displaystyle...

vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may...

complex geometry, the holomorphic tangent bundle of a complex manifold M {\displaystyle M} is the holomorphic analogue of the tangent bundle of a smooth manifold...

Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature...

Hirzebruch–Riemann–Roch theorem applies to any holomorphic vector bundle E on a compact complex manifold X, to calculate the holomorphic Euler characteristic of E in sheaf...

functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental...

connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's...

gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused...

technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations...

tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of...

In mathematics, a Higgs bundle is a pair ( E , φ ) {\displaystyle (E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle...

{\displaystyle (E,\Phi )} where E → X {\displaystyle E\to X} is a holomorphic vector bundle and Φ : E → E ⊗ Ω 1 {\displaystyle \Phi :E\to E\otimes {\boldsymbol...

In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach...

the theory of holomorphic vector bundles (more generally coherent analytic sheaves) on X coincide with that of algebraic vector bundles. Chow's theorem...

canonical bundle is anti-ample Matsusaka's big theorem Divisorial scheme: a scheme admitting an ample family of line bundles Holomorphic vector bundle Kodaira...

applied this new theory vector bundles to develop a notion of slope stability. Define the degree of a holomorphic vector bundle E → ( X , ω ) {\displaystyle...

bundle Ω {\displaystyle \Omega } on V {\displaystyle V} . Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle T...

tangent bundle of a differentiable manifold M {\displaystyle M} is a manifold T M {\displaystyle TM} which assembles all the tangent vectors in M {\displaystyle...

information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under...

same duality statement for X a compact complex manifold and E a holomorphic vector bundle. Here, the Serre duality theorem is a consequence of Hodge theory...

Hermitian metrics on a holomorphic vector bundle. In particular, if the base manifold is Kähler and the vector bundle is its tangent bundle, then the Chern connection...

{\partial }}:\Omega ^{p,q-1}\to \Omega ^{p,q})}}.} If E is a holomorphic vector bundle on a complex manifold X, then one can define likewise a fine resolution...

Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over C P 1 {\displaystyle...

associated bundle E = P × GL ⁡ ( n , C ) C n {\displaystyle E=P\times _{\operatorname {GL} (n,\mathbb {C} )}\mathbb {C} ^{n}} . This is a holomorphic vector bundle...

on vector bundles. The theorem states the following Le Potier (1975): Let X be a n-dimensional compact complex manifold and E a holomorphic vector bundle...

Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial. In particular, every line bundle is trivial, so H 1 ( X , O X ∗ ) = 0...

any noncritical value of a holomorphic map. Smooth complex algebraic varieties are complex manifolds, including: Complex vector spaces. Complex projective...

equations. He was a pioneer in the study of moduli spaces of holomorphic vector bundles on projective varieties. His work is considered the foundation...